Which one of the following is always true with respect to parallel lines ?

Options

A. Sum of their slopes is zero
B. Difference of their slopes is zero
C. Product of their slopes is zero
D. Quotient of their slopes is `-1`

Answer

To determine which statement is always true with respect to parallel lines, we need to analyze the properties of their slopes.

### Step-by-Step Solution:

1. **Understanding Parallel Lines**:
- Parallel lines are lines in a plane that do not meet; they are always the same distance apart and have the same slope.

2. **Defining the Slope**:
- The slope (m) of a line can be defined as the tangent of the angle (θ) that the line makes with the positive x-axis. Mathematically, it is represented as:
\[ m = \tan(\theta) \]

3. **Identifying Slopes of Parallel Lines**:
- If we have two parallel lines, let's denote their slopes as \( m_1 \) and \( m_2 \). Since the lines are parallel, we have:
\[ m_1 = m_2 \]

4. **Evaluating the Options**:
- **Option 1**: Sum of their slopes is zero:
\[ m_1 + m_2 = m + m = 2m \quad \text{(not necessarily zero)} \]
- **Option 2**: Difference of their slopes is zero:
\[ m_1 - m_2 = m - m = 0 \quad \text{(this is true)} \]
- **Option 3**: Product of their slopes is zero:
\[ m_1 \times m_2 = m \times m = m^2 \quad \text{(not necessarily zero)} \]
- **Option 4**: Question of their slope is zero:
- This option is unclear but does not hold true as stated.

5. **Conclusion**:
- The only statement that is always true for parallel lines is that the difference of their slopes is zero. Therefore, the correct answer is:
\[ \text{Option 2: Difference of their slopes is zero.} \]